Periodic solutions

of damped differential systems

with repulsive singular forces

Author:
Meirong Zhang

Journal:
Proc. Amer. Math. Soc. **127** (1999), 401-407

MSC (1991):
Primary 34C15, 34C25

MathSciNet review:
1637460

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We consider the periodic boundary value problem for the singular differential system: where , , and . The singular potential is of repulsive type in the sense that as . Under Habets-Sanchez's strong force condition on at the origin, the existence results, obtained by coincidence degree in this paper, have no restriction on the damping forces . Meanwhile, some quadratic growth of the restoring potentials at infinity is allowed.

**[1]**Antonio Ambrosetti,*Critical points and nonlinear variational problems*, Mém. Soc. Math. France (N.S.)**49**(1992), 139 (English, with French summary). MR**1164129****[2]**Vittorio Coti Zelati,*Dynamical systems with effective-like potentials*, Nonlinear Anal.**12**(1988), no. 2, 209–222. MR**926213**, 10.1016/0362-546X(88)90035-1**[3]**William B. Gordon,*Conservative dynamical systems involving strong forces*, Trans. Amer. Math. Soc.**204**(1975), 113–135. MR**0377983**, 10.1090/S0002-9947-1975-0377983-1**[4]**Patrick Habets and Luis Sanchez,*Periodic solutions of some Liénard equations with singularities*, Proc. Amer. Math. Soc.**109**(1990), no. 4, 1035–1044. MR**1009991**, 10.1090/S0002-9939-1990-1009991-5**[5]**P. Habets and L. Sanchez,*Periodic solutions of dissipative dynamical systems with singular potentials*, Differential Integral Equations**3**(1990), no. 6, 1139–1149. MR**1073063****[6]**A. C. Lazer and S. Solimini,*On periodic solutions of nonlinear differential equations with singularities*, Proc. Amer. Math. Soc.**99**(1987), no. 1, 109–114. MR**866438**, 10.1090/S0002-9939-1987-0866438-7**[7]**P. Majer,*Ljusternik-Schnirel′man theory with local Palais-Smale condition and singular dynamical systems*, Ann. Inst. H. Poincaré Anal. Non Linéaire**8**(1991), no. 5, 459–476 (English, with French summary). MR**1136352****[8]**J. Mawhin,*Topological degree methods in nonlinear boundary value problems*, CBMS Regional Conference Series in Mathematics, vol. 40, American Mathematical Society, Providence, R.I., 1979. Expository lectures from the CBMS Regional Conference held at Harvey Mudd College, Claremont, Calif., June 9–15, 1977. MR**525202****[9]**J. Mawhin,*Topological degree and boundary value problems for nonlinear differential equations*, Topological methods for ordinary differential equations (Montecatini Terme, 1991) Lecture Notes in Math., vol. 1537, Springer, Berlin, 1993, pp. 74–142. MR**1226930**, 10.1007/BFb0085076**[10]**Sergio Solimini,*On forced dynamical systems with a singularity of repulsive type*, Nonlinear Anal.**14**(1990), no. 6, 489–500. MR**1044077**, 10.1016/0362-546X(90)90037-H**[11]**Meirong Zhang,*Periodic solutions of Liénard equations with singular forces of repulsive type*, J. Math. Anal. Appl.**203**(1996), no. 1, 254–269. MR**1412492**, 10.1006/jmaa.1996.0378

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
34C15,
34C25

Retrieve articles in all journals with MSC (1991): 34C15, 34C25

Additional Information

**Meirong Zhang**

Affiliation:
Department of Applied Mathematics, Tsinghua University, Beijing 100084, People’s Republic of China

Email:
mzhang@math.tsinghua.edu.cn

DOI:
https://doi.org/10.1090/S0002-9939-99-05120-5

Keywords:
Singular force,
strong force condition,
damped system,
coincidence degree

Received by editor(s):
September 23, 1996

Additional Notes:
The author is supported by the National Natural Science Foundation of China and the Tsinghua University Education Foundation

Communicated by:
Hal L. Smith

Article copyright:
© Copyright 1999
American Mathematical Society